Not really. As someone who prefers brief comments to in depth explanations to people I barely know I don't see Kirk elaborating further.My general point is that James T. Kirk says what he does in a way that suggests he only knew Pike very casually. SNW shows him hanging around the ship, developing a professional relationship, and probably continuing to do so throughout the series. Which is something very weird to describe, "Yeah, I met him at his promotion to Fleet Captain."
Its technically true but very much against the spirit of what was implied.
SNW Kirk would probably says, "Yeah, I knew him when I was still captain of the Farragut. My brother used to serve on the Enterprise when he was captain."
For starters, Kirk always stands back and seeks to learn information not rattle off useless facts, especially to a superior officer. He also doesn't volunteer more than necessary.
I can see the brief approach being in line with Kirk's character.